Method For Forming Arbitrary Lithographic Wavefronts Using Standard Mask Technology

ABSTRACT

A desired set of diffracted waves using mask features whose transmissions are chosen from a set of supported values are generated. A representation of the mask as a set of polygonal elements is created. Constraints which require that the ratio of the spatial frequencies in the representation take on the amplitude ratios of the desired set of diffracted waves are defined. An optimization algorithm is used to adjust the transmission discontinuities at the edges of the polygons to substantial equality with the discontinuity values allowed by the set of supported transmissions while maintaining the constraints.

TECHNICAL FIELD

The present disclosure generally concerns methods for forminglithographic wavefronts.

BACKGROUND

The well-known “wavefront engineering” approach to improved lithographicperformance is based on the following consideration: At a fundamentallevel, it is often easier to maximize the quality of lithographic imagesby engineering them in the pupil, rather than the object plane. Putdifferently, it is often simpler (from a fundamental point of view) toderive an imaging wavefront that is suitable for producing a highquality image, rather than designing the mask that would actually beneeded to generate the wavefront which forms the image.

We can identify two reasons for this advantage, one conceptual, theother practical.

First, the finite exit-pupil NA is the basic “bottleneck” that actuallylimits the resolution of lithographic images. (Resist diffusion has anon-negligible impact, but resist resolution is almost always finer thanthat of the exposure tool.) Here NA stands for Numerical Aperture, whichis defined as the product of two quantities, namely the sine of thehalf-angular range of the light that is converged to form the image, andthe refractive index of the medium in which the image is formed. Thehighest frequency modulation that the image can contain is given by ½the NA divided by the wavelength. Many practical challenges must beconsidered in state-of-the-art lithography, but the core problem is thatimposed by the limited lens resolution. In order to manage that corechallenge one would like to “push” the most effective wavefront possiblethrough the available NA. (As used herein we will use the term“wavefront” as shorthand for the set of mask spatial frequencies thatare actually collected by a projection lens, e.g. a photolithographiclens, considering all illumination directions present in the source.)Thus, it can be advantageous to work in the pupil domain when trying toobtain the best possible image, particularly in the case of smallcritical cells where intensive optimization is appropriate.

A second advantage of working in the pupil domain is that mask variablesare somewhat inflexible to work with, compared to wavefront variables.For example, shape constraints come into play during direct optimizationof mask variables that are extraneous to the fundamental issue ofmaximizing image quality. These constraints involve the basic topologyof the mask patterns used, along with issues of feasible maskfabrication (e.g. “when edge A is moved out, it cannot be moved closerthan distance d to edge B”). Wavefront variables, on the other hand, arecontinuously adjustable, without mutual constraint. Wavefront variablesare a convenient way to reformulate solutions that are derived from maskpatterns whose shapes are costly to fabricate directly, such asgray-level masks formed with multiple transmission levels to producemulti-level images. Wavefront variables have another convenient aspectwhen periodic boundary conditions are imposed on the object, because insuch cases wavefronts can be completely represented by a specificdiscrete set of diffraction orders, or equivalently by the discreteFourier transform of these orders, and it is these specific orders thatform the image of interest. (Periodic boundary conditions are veryfrequently imposed in lithographic design simulations, either directlybecause the object is truly periodic, or indirectly because thenumerical simulation code uses a discrete grid in the frequency domain.)In contrast, one may not be able to address all intrinsic degrees offreedom in an image by adjusting the positions of available edges in themask, except when the mask edges are so heavily fragmented as to producefar more nominal mask variables than there are true degrees of freedomin the image. That outcome is not assured, and even when all orders canin principle be independently addressed, certain orders may only becoupled very weakly to available edges, depending on the topology of themask design chosen, and this increases the likelihood that extraneousshape constraints will unnecessarily limit the quality of the solutionobtained.

Unfortunately, despite its inherent advantages, lithographic design inthe pupil plane has one significant disadvantage—The known technologydoes not provide a practical method for actually realizing the optimalwavefront, i.e. there is no known method for actually constructing amask using standard photomask technology that will provide a specifiedwavefront as its diffraction pattern. The issue of practicality is keyhere—One can, of course, find a mathematically valid mask solution bytaking the Fourier transform of the desired wavefront (after choosingsome nominally arbitrary [but actually consequential] shape for theuncollected portion of the wavefront); however, this will produce a“mask” that is continuously varying, and so not manufacturable.Producing a specified wavefront with a manufacturable mask is anon-trivial problem.

Manufacturable mask features must take the form of openings in abackground film, and these openings must be fairly coarse in size(though they can be smaller [when scaled to “1×”] than the minimum-sizedfeatures that can actually be developed in resist; also, the perimetersof mask features can contain fine jogs that are smaller than thesmallest mask features). Another limitation is that the transmission ofeach mask opening is, in the simplest instance, fixed at thetransmission level of the substrate. Modern masks allow slightly moreflexibility than this, but in general feature transmission should bechosen from one or two allowed values (in addition to the backgroundtransmission, which may be nonzero), i.e. masks must generally be binaryor trinary in order to meet production-grade feature placementspecifications, and to contain fabrication cost. For example, in aso-called Levenson mask, the intensity transmission in any region canonly be 0 or 100%, and the transmitted phase can only be 0° or 180°. Ingeneral, restriction of the phase shift to 0° or 180° causes thetransmission to be real-valued, and the resulting pure-real character ofthe transmitted wavefront causes critical dimensions in the image tohave better stability through focus. For this reason practical maskfilms conventionally have a transmission phase of either 0° or 180°.So-called grey-level masks whose features have more than two differentintensity transmissions generally cannot meet practical featureplacement requirements.

Critical features in manufacturable masks must nominally be polygonal,i.e. they must be designed with straight edges (though the limitedresolution of mask writing technology will cause significant cornerrounding). Also, critical features must usually be “Manhattan”, i.e.their edges can only take right-angle turns, with the edges of alldifferent features being parallel or perpendicular to one another.(However a limited number of features with non-Manhattan edgeorientation is sometimes acceptable, such as features with 45°orientation.)

The finite thickness of the patterned mask films poses another practicalproblem for mask design, since it causes the transmission to locallydeviate from its nominal value, particularly in the vicinity of thefeature edge. More specifically, the light transmitted through maskapertures will only match the transmission of the mask blank atpositions that are somewhat removed from the aperture edge, and likewisethe transmission in unopened regions will deviate from the transmissionof the background films at positions that are adjacent to apertureedges. The transmission discontinuity arising at the verticaltopographic edges of features will therefore not match the nominaldiscontinuity as defined by the separation between the basictransmission values supported by the mask technology. Such deviationsfrom the nominal behavior are due to the interaction of theElectromagnetic fields with the complex topography of the patterned maskfilms; these deviations are referred to as “EMF” (for Electro MagneticField) effects. Roughly speaking, we can regard EMF effects as being aconsequence of the finite thickness of the physical films or trenchesthat are etched out to form the features that are written on the mask.EMF effects usually become more significant as the film thicknessbecomes relatively larger in comparison to the feature widths andwavelength. Mask films are very roughly of order 70-100 nm in thickness,and printed features have until recently been larger than the exposingwavelength (which today is typically 193 nm). Since lithographic masksare usually 4× enlarged, it has thus been reasonably accurate to neglecttheir topography, and treat them as ideal two dimensional (2D) masks(the so-called Thin-Mask Approximation, or “TMA”). Even today, itremains true that the basic lowest order behavior of lithographic masksis generally captured by the TMA approximation. However, while EMFeffects can usually be regarded as a perturbation on the TMA behavior,the significance of their impact can be quite substantial in the contextof the stringent tolerances of photolithography.

As shown in FIG. 1, the finite thickness of mask topography causesperturbations in the transmitted field. The transmission from pointsthat are appreciably distant from the topographic edge is littlechanged, but the perturbation can become non-negligible near theperimeter of mask apertures, particularly at the small feature sizescharacteristic of modern masks. To lowest order, the in-phase (realvalued) transmission change is roughly that produced by a smallextension or retraction of the associated edge.

As shown in FIG. 2, the thin mask model is usually able to capture thegross behavior of lithographic images; in this example the printed imagesize is predicted within ˜11.5%. (Feature size is 50 nm at all plottedperiodicities.) The prediction error becomes less than 2% if theabsorber edges in the Thin Mask Approximation (TMA) model are extended(biased) from the edge in a way that mimics the topography-inducedtransmission change.

The projection lens is incapable of resolving the fine structure of theEMF-induced discontinuity in the fields, and it is known (J.Tirapu-Azpiroz and E. Yablonovitch, “Incorporating mask topography edgediffraction in photolithography simulations,” J. Opt. Soc. Am. A 23,4(2006): p. 821) that EMF effects can be approximately reproduced using aTMA model in which the edge fields are rendered as small strip-likefeatures of essentially fixed transmission (generally a complextransmission) that are assumed for simulation purposes to lie along theaperture boundaries. More precisely, since these perturbing strips(known as boundary layers) are considerably narrower than the lensresolution, their width can (in first approximation) be modestlyre-adjusted as long as a compensating adjustment is made in theirtransmission, holding the width-transmission product effectivelyconstant. (We qualify this as “effectively” constant because we requirethat the width-transmission product include the thin-mask transmissionthat would otherwise have been present in the strip of mask-area thatthe boundary layer displaces.) When the boundary layer is scaled to havea transmission of order unity in magnitude, its width will usually bevery roughly of order λ/20, i.e. boundary layers are usually stronglysub-resolution.

Since boundary layers are unresolved, the in-phase part of their imagecontribution is very similar to that which would be obtained byrecessing the aperture edge by a distance that would deliver a matchingamplitude contribution (or extending the edge to appropriately occludethe illumination, depending on the sign) in the form of a simple bias.

It is known that the impact on transmitted amplitude EMF effects can tofirst order approximation be corrected by simple biasing, in order tocarry out mask design in the basic mode known as Optical ProximityCorrection (“OPC”); see FIGS. 1 and 2. OPC involves adjusting theposition of the topographic edges of mask features in such a way thatthe contour of the printed image falls at a specified position.Essentially, the EMF-induced incremental change in delivered intensityat the feature edge causes a change in the contour position, and themask aperture must be biased in the opposite direction to undo theshift. In many cases the simple opaque bias model allows the intensitychange to be calculated both accurately and rapidly, making OPCcorrection with topographic masks possible.

However, advanced forms of lithographic optimization that aim to printat the extreme limits of resolution must worry about the processrobustness of the printed image, and focus sensitivity is a criticalaspect of process robustness. Focus sensitivity is impacted by the phaseof the transmitted light, and the in-quadrature component of thevertical edge field perturbation cannot be compensated by a shift inedge position (as shown in FIG. 6). As a result, it is only possible tocompensate the degradation in focus robustness that EMF induces in anaveraged way when shape adjustment is employed as the compensationmethod. The in-quadrature (or imaginary) component of the EMFperturbation can therefore be considered more critical than the in-phase(or real) part, and the magnitude of the in-quadrature component islargely a function of the mask topography, which in turn depends on thephase and transmission that are chosen for the mask aperture andbackground regions.

As shown in FIGS. 3A, 3B and 5, the main impact of the in-quadrature(imaginary valued) component of EMF-induced image changes is a pitchdependent focus shift. The shift of plane of best focus with featuresize degrades the common window of the process or “common processwindow”. The term “Common PW” is short for common process window, andrefers to the range of fluctuations in dose and focus over which thefluctuations in a lithographic image remain within tolerance.

As shown in FIGS. 4A-4B, the approximate boundary layer model of EMFeffects provides a reasonably accurate calculation of thefeature-dependent shifts in focus that are produced by mask topography,with broadly accurate results being obtained down to quite small featuresizes.

In many cases the wavefronts which produce the best-performing imagescan only be created from masks which have transmitting regions of both0° and 180° phase, since the availability of both polarities makes iteasier to form adjacent bright areas of the image with fields ofopposite sign, creating a high contrast dark fringe between the brightfeatures where the field passes through zero amplitude as it changessign. Such opposite phases can also be produced using the tilt-phasethat is generated with off-axis illumination, but this is less flexiblethan deploying phase-shift on the mask when complex patterns areinvolved. Unfortunately, topography effects make it hard to maintain thebenefits of phase shift imaging as the dimensions of mask featuresshrink. EMF effects increase as topographic-edge-regions occupy anincreasingly large portion of the mask area, and the three-dimensional(3D) topographic step that is present between regions that arephase-shifted tends to be relatively large. As noted above, the field inthe vicinity of the step exhibits a phase that is different from the 0°and 180° phases that are attained in the extended open areas on eitherside of the step. These latter nominal transmittances are pure real(in-phase) even though phase shifters have been employed, but themagnitude of the imaginary (in-quadrature) component that EMF effectsinduce at vertical topographic edges will tend to be larger with therelatively thick films that phase-shift masks typically employ. Thislocalized quadrature component can cause focus shifts even for opaquebinary masks, and in general the miss-phased field will occupy a largerfraction of the transmitted beam when features are small. And as we haveseen, this quadrature error also makes it impossible to fully correctthe impact of EMF by pure shape adjustment alone.

FIG. 5 shows focal behavior of printed features when a known mask offinite thickness topography is used. Images from TMA masks have adesirable zone of focal stability that is centered at z=0, since thederivative of image intensity with respect to z will be zero at thatfocus (assuming the usual symmetric source). However, when the thicknessof mask topography is non-negligible, one sees from plots like these offeature size vs focus (so-called Bossung curves) that the positions ofbest focus (center of the regions of focal stability) are shifted awayfrom z=0 in a non-uniform, feature-dependent way.

As shown in FIGS. 3A-3B, biasing cannot correct focal shifts that arecaused by the quadrature component of the EMF perturbation. At a fixedfocus position, a TMA calculation using a biased mask is incapable ofreproducing the true topographic EMF behavior through the full doserange.

The known technology provides only limited means for dealing with thesepractical difficulties of wavefront engineering. Consider first thelimited flexibility that adjustment of conventional mask shapesprovides, and the inability of such adjustments to easily address alldegrees of freedom in the image. If one is willing to set aside issuesof mask manufacturability, there is a known method for optimization oflithographic images that operates in the mask plane, while managing tocapture much of the flexibility of wavefront design; this is the methodof image optimization using high density bitmap masks, in which everypixel is independently adjustable, and where the pixels are so small asto provide effectively continuous addressability of the mask. Bitmapmasks provide the flexibility needed to achieve optimal images, but theycontain far more variables than necessary (which severely slows mostoptimization algorithms). Also, bitmap masks are not practicallymanufacturable. State-of-the-art mask technology typically requires thatisolated mask openings (e.g. bitmap pixels in the case of bitmap masks)be sized larger than perhaps ¼ the width of the smallest feature thatcan actually be resolved (i.e. printed) in a single wafer image (exceptscaled up by the lens magnification). The edges of mask features cancontain jogs that are much finer than this, but small jog-like serifs donot remove the practical difficulty in fabricating bitmap masks, for thefollowing reason: Since bitmap pixels represent a large number ofindependent variables, they will be highly redundant, hence many of thepixel adjustments that improve the objective function are likely to bespatially isolated from other pixels of the same polarity as theparticular pixel that is actually adjusted at any given step, and theresulting small isolated pixel apertures are not manufacturable.

This lack of contiguity can be circumvented when the problem is linear,but mask optimization problems are inherently quadratic (at best), sincethe exposing intensity is a quadratic function of diffracted amplitude.Shape constraints can be included in the optimization procedure toinhibit the use of isolated pixels, but then the algorithm becomes boundonce again by topological constraints that are irrelevant to the imagingprocess itself (where the working solution should be able to representany imaging wavefront that can be propagated through the bandlimitinglens NA), and in addition the working solution can fall into extraneouslocal minima that involve non-essential topological constraints arisingfrom happenstance clustering. Often these manufacturability requirementsare addressed by adding penalty terms to the objective function, butperformance is then penalized when the objective is re-weighted toemphasize manufacturability, and in addition the manufacturabilityrequirements are often incompletely satisfied.

Though lithographic design in the pupil plane has been known for manyyears (e.g. under the rubric of “wavefront engineering”), the abovedisconnect from mask fabrication has generally restricted wavefrontengineering to the role of conceptual aid, rather than full workingprocedure. One-dimensional patterns are a partial exception to this;known methods for laying out one dimensional (1D) assist featuresprovide a fairly complete link between the desired 1D diffractionpatterns and feasible masks. Smith (B. W. Smith, “Mutually OptimizingResolution Enhancement Techniques: Illumination, APSM, Assist FeatureOPC, and Gray Bars”, SPIE v.4346—Optical Microlithography XIV, (2001):p. 471) provides a discussion of pupil-plane optimization and theassociated determination of suitable 1D masks.

However, it would be desirable to have a method for producing anarbitrary wavefront within the lens exit pupil, without being restrictedto 1D. Such a method could in principle be used to produce any imagethat a given litho exposure tool is theoretically capable of. Thisincludes images that have been designed using wavefront variables, aswell as images which known lithographic methods could only produce usingidealized masks whose fabrication would be impractical, such as imagesfrom non-manufacturable gray-level masks that employ more than twointensity transmission levels, or images from masks that containnon-manufacturable aperture shapes. Such a method could in additionproduce images that are initially designed using impractical idealizedmask solutions, and then further refined using wavefront variables. Ingeneral, problems of practical mask fabrication would be separated fromthe core problem of determining the best possible image.

Rosenbluth et al. took an important step towards such a capability withan algorithm described in A. E. Rosenbluth et al., “Optimum Mask andSource Patterns to Print a Given Shape,” JM3, 1, 1 (2002), p. 13. Thisreference shows how to devise a binary or trinary mask that willreproduce a specified diffraction pattern by solving a single linearprogramming (LP) problem. Mask features provided by this LP will usuallytake the form of reasonably large contiguous mask openings, rather thanthe tiny isolated halftones of bitmap masks. (It should be noted thatwhile the features in the LP solution are usually of practical size,they can also include unrealistically fine “tendrils”, which in theRosenbluth et al. method are essentially removed by manualintervention.)

However, a drawback to this known method is that the features providedare very far from Manhattan—Feature edges not only have arbitraryorientation, but are actually curved in complex ways. FIG. 7 shows anexample, namely a binary mask (transmission=±1) that produces anoptimized diffraction pattern for a dynamic random access memory (DRAM)isolation level (see FIG. 8), generated using a known method. Width ofcell is about 3 λ/NA, height about 1.5 λ/NA, with λ=248 nm, NA=0.68.Unfortunately, these curved mask geometries are not manufacturable, dueboth to lack of Manhattan (or even polygonal) apertures, and thepresence of a few overly fine connections between the generallycontiguous apertures.

It is possible with some trial and error to semi-manually derive aManhattan layout from masks produced by this algorithm (e.g. the abovepaper by Rosenbluth et al. shows a Manhattan mask that is semi-manuallyderived from the FIG. 7 solution). To do so one draws on the plottedmask a staircased line that approximately follows the perimeter of eachmask region. One then reads the coordinates of the staircase cornersfrom the plot, and enters them into an optimization program whichattempts to reproduce the desired diffraction orders by adjusting thecorner positions. Convergence is very fast if the staircasing is fine,but the masks then become more difficult to fabricate. On the otherhand, the corner optimizer typically fails to converge when thestaircasing is coarse. Usually one can find an acceptable compromiseafter a bit of trial and error.

However, this method is far from ideal. First, the final mask featuresusually contain a large number of difficult-to-fabricate jogs andserifs, i.e. protruding features with aspect ratio of order 1 that havetwo or more edges with length near the limit of fabricability. Fragmentsthat protrude only slightly from a long edge (i.e. having aspect ratiosfar from 1) are not a significant concern, nor arenear-unit-aspect-ratio structures that are relatively large. A limitednumber of more difficult jogs (of small but acceptable size, and compactaspect ratio) can be handled, and these jogs can be quite a bit smallerthan the minimum allowable isolated mask feature (i.e. it is acceptableto have small jogs that merely adjust the perimeter of a larger, fullyresolved feature.)

FIG. 8 shows a DRAM isolation pattern used as an example to explain thepresent method. Rectangles should be printed as dark. Periodicity ofrectangular optical unit cell is 1120 nm in the x direction, 560 nm inthe y direction.

Unfortunately, a hand-staircased solution often contains more such jogsthan is desirable, and also more jogs than are fundamentally necessaryto reproduce the diffraction pattern. Another disadvantage to thehand-staircasing method is simply that it is a manual procedure, and sois time-consuming and prone to error. Also, very similar patterns may bestaircased in appreciably different ways if the human engineer involveddoes not recognize or recall previously handled cases. Ideally thiswould not matter since all solutions will nominally produce the sameimage; however in practice this would tend to increase variation inCritical Dimensions (CD's) across the printed chip level.

SUMMARY OF THE INVENTION

There is disclosed a method for forming arbitrary lithographicwavefronts using standard mask technology. Optimization can be used toobtain a manufacturable mask that will diffract a specified wavefront,but the criteria for manufacturability are sufficiently complex andnonlinear as to require local optimization. It is then necessary to finda starting design that provides very nearly the correct wavefront usingshapes that can be made manufacturable without breaking the initialtopology, since local optimization involves smooth and continuousadjustments. It is this starting design that allows the localoptimization to avoid being compromised by extraneous topologicalconstraints. Such an approximately manufacturable starting mask can bedesigned using a Manhattan grid that has variable spacings. The meangrid spacing is chosen to correspond roughly to a typical fragment size;more specifically, the mean grid spacing is chosen (using formulassupplied below) to be sufficiently fine that the specified wavefront canbe reproduced, yet sufficiently coarse that the mask is approximatelymanufacturable. The specific gridline positions can depart from theaverage spacing, and these positions are adjusted in a way thatconverges to manufacturability; more specifically, the gridlines arepositioned to permit as large a nominal discontinuity in masktransmission as possible across gridlines (using the method of the nextparagraph), eventually being adjusted to the point that every nominaldiscontinuity is as large as one of the allowed discontinuities definedby the differences between the binary or trinary set of allowedtransmission values supported in the mask manufacturing process. Thepresent method makes additional adjustments to account for thediscontinuities arising from finite thickness topography, but thegridline adjustments involve only the nominal discontinuities in thethin-mask transmission. Additional shape adjustments are made tocompensate the real (in-phase) part of the EMF discontinuity, andadditional adjustments to the mask topography are made to compensate thequadrature discontinuity.

The gridline adjustment can be accomplished in two basic ways, each ofwhich increases the average (non-zero) nominal discontinuity acrossgridlines, with the important qualification that gridline sectionsacross which there is no discontinuity are not counted, i.e.discontinuities are either flattened down to zero, or increased to alevel consistent with the mask technology, thereby removing intermediatetransitional transmissions. In a preferred embodiment, the presentmethod begins by applying the first of these methods, which is to setthe transmission of the blocks (i.e. rectangles) between adjacentgridlines to those particular values which maximize diffracted intensity(some of these initial transmission values not being manufacturable,since the required wavefront should be achieved precisely). It will beshown below that this drives a majority of the rectangles to one of theextreme transmission values allowed by the mask technology, and that,among the rectangles in this category, those having the sametransmission tend to cluster together. These clustered rectangles haveno transmission discontinuity where they join within the interior of theclusters; however, somewhat larger discontinuities are present along theborders of the clusters. Next, the present method employs the secondmethod for increasing the average (non-zero) discontinuity acrossgridlines, which is to insert new gridlines through the particular rowsor columns of blocks in which the transmission of a large number ofblocks had to be kept far from the transmission extremes supported bythe mask technology in order to reproduce the specified wavefront (i.e.gridlines are inserted through rows and columns with a large number of“graylevel blocks”, which are present because of the need to preciselytune the diffracted spectrum to the correct amplitudes). Gridlineinsertion in effect replaces each such graylevel block by a pair ofblocks. The initial gridline separation is chosen to be smaller than thelens resolution, which means that each newly formed pair of blocks canhave almost the same optical impact as the original graylevel parentblock even when the daughter blocks are given (opposing) non-grayleveltransmissions, as long as the relative area of the daughter blocks isset in the proper proportion. Both daughter blocks can then be givenmanufacturable transmittances. However, in a preferred embodiment, thisrelative area adjustment is not made right away. Instead, the diffractedintensity is re-maximized with the new gridlines in place, and thetransmission of all blocks that remain at unsupported values are roundedto the nearest supported transmission level, and finally the positionsof all gridlines are adjusted to remove any inaccuracies that wereintroduced in the desired spectrum by the rounding step. The steps ofthis procedure are summarized in FIG. 10.

For example, FIG. 9 shows a diffraction pattern that has been optimizedto print the FIG. 8 isolation pattern. To explain the present method thetext shows how a practical mask can be designed to produce thiswavefront.

FIG. 10 shows a flowchart summarizing a preferred embodiment of the maskdesign algorithm. EMF correction is applied in a later step.

FIG. 11 shows an optimized source for mask that produces the FIG. 9wavefront. The pupil fill parameter σ_(M) is 0.84.

In another aspect, there is described a memory storing a program ofcomputer readable instructions executable by a processor to performactions directed to generating a desired set of diffracted waves usingfeatures of a lithographic mask for which a set of supportedtransmissions are chosen from a set of supported values, the actionscomprising: creating a representation of the mask as a set of polygonalelements, defining constraints which require that the ratio of thespatial frequencies in the representation take on the amplitude ratiosof the desired set of diffracted waves, using an optimization algorithmto adjust the transmission discontinuities at edges of the polygonalelements to substantial equality with the discontinuity values allowedby the set of supported transmissions while maintaining the constraints.

In one aspect of the memory, the optimization algorithm comprisesiterated steps, the iterated steps comprising: forming a 3Dtopographical representation from the polygonal elements, and simulatingit with a full-3D Maxwell solver to calculate the Fourier transform ofthe edge discontinuities.

In another aspect of the memory, the iterated steps further comprise:calculating a compensating adjustment that cancels the deviations of theFourier transforms of the edge discontinuities from the required spatialfrequency ratios.

In a further aspect of the memory the iterated steps further comprise:forming an adjusted set of Fourier orders using the compensated edgeFourier transforms calculated in the previous step and use them togenerate with thin-mask wavefront engineering a new set of iteratedpolygonal elements.

In another aspect of the memory, the optimization algorithm furthercomprises determining the iterations when the Fourier transform of the3D topographical representation of the iterated polygonal elementssubstantially reproduces the amplitude ratios of the desired set ofdiffracted waves.

In a yet further aspect of the memory, one or more transmissiondiscontinuities are driven to substantial equality with an allowed valueby: forming the 3D topographical representation of the polygonalelements, calculating the transmission discontinuity at the edges of thepolygonal elements, and adding features to the mask whose in-quadraturetransmission component substantially cancels the in-quadrature componentof the transmission discontinuities at edges of the polygonal elements.

In still yet another aspect of the memory, one or more transmissiondiscontinuities are driven to substantial equality with an allowed valueby: giving the desired ratios of spatial frequencies complex values thatprovide the image with a desired behavior through focus, forming the 3Dtopographical representation of the polygon elements, calculating thetransmission discontinuity at the edges of the polygonal elements, andadding features to the mask whose quadrature transmission componentcombined with the quadrature component of the transmissiondiscontinuities at the edges of the polygonal elements provides thein-quadrature part of the complex values of the desired spatialfrequency ratios.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other aspects of these teachings are made more evidentin the following Detailed Description, when read in conjunction with theattached Drawing Figures, wherein:

FIG. 1 depicts a finite thickness of mask topography causingperturbations in the transmitted field.

FIG. 2 depicts the impact of simple biasing in a thin mask model.

FIGS. 3A-3B depict the main impact of the in-quadrature component ofEMF-induced image changes as being a pitch dependent focus shift.

FIGS. 4A-4B depict the accuracy of an approximate boundary layer modelof EMF effects to estimate focus shift produced by in-quadrature EMFeffects on the mask topography.

FIG. 5 shows focal behavior of printed features when a known mask offinite thickness topography is used.

FIG. 6 depicts the inability of biasing to correct focal shifts that arecaused by the quadrature component of the EMF perturbation.

FIG. 7 shows a binary mask that produces an optimized diffractionpattern for a DRAM isolation level.

FIG. 8 shows a DRAM isolation pattern used as an example to explain thepresent method.

FIG. 9 shows a diffraction pattern that has been optimized to print theFIG. 8 isolation pattern.

FIG. 10 shows a flowchart summarizing a preferred embodiment of the maskdesign algorithm.

FIG. 11 shows an optimized source for mask that produces the FIG. 9wavefront.

FIG. 12 shows a working graylevel mask design for FIG. 8 isolationpattern.

FIG. 13 shows one octant of the FIG. 12 mask, along the top edge, to theright of the centerline.

FIGS. 14-22 show various results of sample solutions that illustratevarious aspects of the present method.

FIG. 23 shows a brick pattern used to illustrate the process of EMFcorrection according to an embodiment of the present method.

FIG. 24 shows EMF and TMA images of the FIG. 23 mask.

FIG. 25 shows an EMF-corrected version of the FIG. 23 mask.

FIG. 26 plots the image of the FIG. 25 mask as calculated using arigorous Maxwell solver.

FIG. 27 shows basic anti-Boundary Layer (“BL”) configurations.

FIG. 28 shows that with a chromeless (“CPL”) mask a single additionalpatterning step can create anti-BL's that are centered on the nominaledges.

FIG. 29 shows correction of the EMF-induced feature-dependent focalshifts seen in FIG. 5 using anti-boundary-layers.

FIG. 30 shows correction of the EMF-induced focal shifts seen in FIG. 6using a negative anti-boundary-layer.

FIGS. 31A-31B show anti-BL's to correct focus variations in printing 50nm lines at different pitches.

FIGS. 32A-32D show anti-BL correction of a chromeless phase-shift mask(a so-called CPL mask) for a test problem.

FIGS. 33 and FIGS. 34A-34B show evidence of electromagnetic phase errorscausation.

FIG. 35 shows an isotropic boundary layer.

FIG. 36 shows EMF correction on the reticle.

FIGS. 37-40 show antiBL optimization.

FIG. 41 shows Thin Mask Approximation results for the photomask.

FIG. 42 shows diffraction orders diffracted by the photomask and thephotomask electromagnetic near fields.

FIG. 43 shows a computer configured with an optimization algorithmstored in its memory that is suitable for the adjusting transmissiondiscontinuities according to an exemplary embodiment of the invention.

DETAILED DESCRIPTION

We now explain the procedure summarized above in more detail, using asan example a DRAM pattern that is to be printed at NA=0.68 nm and k=248nm (FIG. 8). The methods of Rosenbluth et al. show that this pattern isprinted with optimum process window if the optimized source of FIG. 11is used together with the optimized mask wavefront of FIG. 9. The known(non-manufacturable) FIG. 7 mask successfully produces this optimizedwavefront; we now show how the present method can produce the samewavefront using a mask that is manufacturable.

The first step of the method is to lay out a set of Manhattan gridlineswith a uniform (coarse) spacing. (Later the spacings are made to vary.)The initial spacing should very roughly match that expected for typicalfragment-lengths in the final mask, and should be at least 2 or 4 timesthe minimum allowable fragmentation length. The symbol α is used todenote the average separation between gridlines, expressed in units ofλ/NA. Appropriate choices will vary with the mask technology beingpracticed; reasonable values are e.g. 30 nm or 0.15 λ/NA, as we nowexplain in more detail.

It is convenient, though not essential, to round the initial α in such away that the initial gridline separation evenly divides the mask regionunder consideration. The grid size must be chosen smaller than 0.25 λ/NAto ensure that the desired diffraction pattern can be created, but useof excessively small fragments will make the mask harder to fabricate,e.g. one should usually choose α greater than or not appreciably smallerthan 0.1.

On the other hand, choosing α very close to the sampling-theorem limitwill slow convergence, particularly when the illumination has asignificant spread off-axis, i.e. one should usually choose e.g. α istypically less than or not appreciably larger than 0.2.

We will return below to the considerations involved in choosing theinitial value of α, where we show that α values in this range aresuitable for efficiently forcing (after just one iteration) a majorityof transmission discontinuities across gridlines in the working mask totake on supported values (including zero). For our demonstration problemwe will choose an initial grid spacing of 0.15 λ/NA≅55 nm.

It should also be noted that our preferred embodiment, which is based ongridlines that span the entire field, is not ideally suited to verylarge cells. The method can be modified to use e.g. four separate blocksof gridlines each covering one quadrant of the field. This might beappropriate if 100 λ/NA is typically less than or not appreciably largerthan P_(x)P_(y).

For good understanding, PxPy is the product of the cell size along the xaxis and the cell size along the y axis. For such large fields thenonlinearities of the method will also slow down the solution, and itcan be preferable to solve the problem separately in each of a number ofsubdivisions of the original field, and then stitch the differentsolutions together. This can be done whenever the original field size isappreciably larger than the lens resolution (see below).

For symmetric patterns like our demonstration DRAM cell, each rectanglecontributes to the {n,m} Fourier order an amplitude proportional to:

$\begin{matrix}{{\Delta \; {A_{i,j}\left( {n,m} \right)}} \propto {{\left\lbrack {x_{i + 1} - x_{i}} \right\rbrack \left\lbrack {y_{j + 1} - y_{j}} \right\rbrack}\sin \; {c\left( \frac{\pi \; {n\left\lbrack {x_{i + 1} - x_{i}} \right\rbrack}}{P_{x}} \right)}\sin \; c{\quad{{\left( \frac{\pi \; {n\left\lbrack {y_{j + 1} - y_{j}} \right\rbrack}}{P_{y}} \right) \times {\cos \left( \frac{\pi \; {n\left\lbrack {x_{i + 1} + x_{i}} \right\rbrack}}{P_{x}} \right)}{\cos \left( \frac{\pi \; {n\left\lbrack {y_{j + 1} + y_{j}} \right\rbrack}}{P_{y}} \right)}},}}}} & \lbrack 1\rbrack\end{matrix}$

where the rectangle falls between x gridlines i and i+1, and y gridlinesj and j+1 (referred to as “rectangle [i,j]”). General nonsymmetricpatterns can be handled using the polygon Fourier transform formulasgiven in C. P. Ausschnitt, R. L. Gordon, C. J. Progler, and A. E.Rosenbluth, “Integrated lithographic print and detection model foroptical CD,” U.S. Pat. No. 6,869,739 (2005); see also R. L. Gordon andA. E. Rosenbluth, “Lithographic simulations for the 21st century with19th Century Tools,” in SPE v.5182: Wave-Optical Systems Engineering II,ed. Frank Wyrowski (2003), p. 73.

P_(x) and P_(y) are the x and y periodicities of the overall field. Notethat under the staggered symmetry of this DRAM layout, the rectangularoptical unit cell has twice the area of the true diamond-shaped“crystallographic” period, which is stepped out along diagonal basisvectors. This doubling, together with the pattern's bilateral symmetryacross the x and y axes, means that one octant of the optical cell issufficient to define the remainder of the pattern. Note that in treatingthe field as periodic we are implicitly making an assumption here thatthe exit pupil wavefront has been specified by tabulating its value at adiscrete set of points (on a 2D grid). Such a discrete tabulationamounts to an automatic assumption of nominal periodicity, and iscommonly made in lithographic simulations. On the other hand, if thewavefront is continuous, the scale of its most rapid variations willdefine a maximum image field size (which would usually be knownindependently from the nature of the problem). The mask region can thenbe made periodic by adding a small buffer zone (guard band) to thisbounded field (scaled to the object plane) and then digitally samplingthe wavefront; such a step is equivalent to sampling the wavefront on agrid that is finer than the most rapid wavefront variations. When thetrue circuit pattern is not periodic one must ensure that the guard bandis larger than the lens resolution.

In cases where the wavefront is specified for a very large field it maybe faster to divide the mask field into separate sections whose featuresare solved for separately, and then stitched together. To isolate asmall mask region in the wavefront one convolves the full wavefront withan appropriate sinc function (to blank out the image field outside theregion of interest, allowing for a small buffer zone), and thenresamples the wavefront with the appropriate coarse grid. A point toconsider here is that the FIG. 10 algorithm in its standard form willmaximize the intensity of each mask region, opening up the possibilityof a dose mismatch after stitching. One way to handle this is to add apostprocessing step in which the gridlines or feature edges in eachsegment are readjusted to match the intensity (e.g. in the referenceorder) of the dimmest mask section obtained. If this is contemplated onemight reduce parameter α slightly to provide additional degrees offreedom. Another way to match doses during stitching is to include fixedanchoring patterns at the boundaries of the variable grid when thealgorithm is started on a particular section. This is done after aninitial run to determine the dimmest mask section. The other masksections are then rerun (with slightly larger buffer zones) beginningwith sections adjacent to the dimmest section. Border features from thedimmest section, of size larger than the lens resolution, are includedas anchoring features when the adjacent sections are solved, but theseanchor features are not adjusted. Since the FIG. 10 algorithm will thenprovide the correct wavefront shape from regions that include anchoredareas (of fixed contents) that are larger than the lens resolution, thealgorithm must also provide the correct intensity in the wavefront,since the adjustable part of the mask field is too far away to provideadditional light to the entire anchored area of the image, whoseintensity is thus fixed, assuming that an exact match is made to thediffraction orders of the full field.

These stitching methods (gridline adjustment, edge adjustment, oranchoring features) can also be used to match doses from mask regionsthat are not adjacent. In such cases the anchoring features must bespaced away from other features by a distance larger than the lensresolution.

Eq. [1] is written as a proportionality because it does not include theactual transmission assigned to the rectangle. The total amplitude in aparticular order {n,m} is given by the sum of the transmissions assignedto each rectangle when weighted by the eq. [1] coefficients:

$\begin{matrix}{{{{A\left( {n,m} \right)} - {A_{back}\left( {n,m} \right)}} = {\sum\limits_{i,j}^{\;}{\left( {t_{i,j} - t_{back}} \right)\Delta \; {A_{i,j}\left( {n,m} \right)}}}},} & \lbrack 2\rbrack\end{matrix}$

where t_(i,j) is the amplitude of rectangle [i,j], t_(back) is thebackground transmission, and A_(back) is the fixed background amplitude.Thus, the Fourier amplitude is given by an expression that is linear inthe mask amplitudes.

The FIG. 9 prescription defines the desired diffraction pattern for oursample problem using the {n=2, m=0} diffraction order as a normalizingreference. Thus, when the linear Fourier amplitude expression for the{2,0} order is multiplied by any numerical entry from the right columnof FIG. 9, the result must match the Fourier amplitude expression forthe corresponding order in the left column, i.e.

A(n,m)=N _(n,m) A(r _(x) ,r _(y)),)   [3]

where N_(n,m) is the normalized target amplitude for order {n,m}, and{r_(x),r_(y)} designates the reference order, {2,0} in our example.

According to eqs. [2] and [3] our requirement that the mask produce therequired diffraction pattern can be expressed as a list of constraintsthat are linear in the rectangle transmission values. The rectangletransmissions must also be constrained to lie within the range availablein the mask technology of interest. One cannot expect, however, thatwith our initial choice of gridline positions it will be possible toachieve the desired order amplitudes using only the discretetransmission values that are allowed.

We can force as many rectangles as possible to the extreme allowedtransmission values by maximizing the intensity of the diffractionpattern (while preserving its shape as specified in FIG. 9). Thoughintensity is quadratic in the mask variables, we are fixing the shape ofthe diffracted wavefront, hence intensity is maximized whenever apositive-specified order amplitude is maximized, or when anegative-specified amplitude is minimized. (Complex amplitudes can behandled by e.g. maximizing the real part of an order amplitude when thereal part is specified to be a positive number relative to the referenceorder, with similar rules for negative or imaginary parts.) Thus, in theFIG. 9 example, the intensity of the diffracted wavefront can bemaximized by minimizing the amplitudes of the {0,0}, {1,1}, {3,1}, or{4,0} orders, or by maximizing the amplitudes of the {0,2}, {2,0}, {2,2}or {5,1} orders.

Thus, if we had the luxury of being allowed to individually set thetransmission of mask rectangles to arbitrary gray levels between thesupported extremes (e.g. between −1 and +1 in a chromeless or Levensonmask, or between −(0.06)^(0.5) and +1 in an attenuated phase-shiftmask), the problem of maximizing diffracted intensity while achievingthe desired wavefront shape would be a linear one. In other words,because our constraints on wavefront shape allow a linear amplitudeobjective to be substituted for our quadratic intensity objective, andsince these constraints are themselves linear (as are the constraintslimiting transmission to the range supported by the mask technology), wecould treat the entire problem as linear if graylevel blocks wereallowed. This in turn would mean that the problem could be solvedglobally (and rapidly) by standard linear programming routines (and afeasible solution can be obtained by simple Fourier transform ifcontinuous variation were manufacturable). Unfortunately, arbitrarytransmittances aren't supported in practical lithographic masktechnologies, where each transmittance level other than that ofbackground requires the patterning of a separate film which mustdedicated to that gray level.

However, we can force the majority of the rectangles to havetransmittance matching one of the supported limits by properly choosinga when we maximize the amplitude of one of the orders (i.e. as explainedabove, an order whose normalized amplitude is positive relative to thereference, or when we minimize the amplitude of a negative-specifiedorder). The underlying reason for this forced conformance is that theamplitude is in itself an unbounded function of the rectangletransmissions—So long as each rectangle has finite area, increasing ordecreasing its transmission will monotonically either increase ordecrease the amplitude of the chosen diffraction order. It is only theconstraints of the problem that prevent further increases after asolution is reached.

Thus, when we find a solution to our problem of maximizing diffractedintensity by maximizing or minimizing a particular order chosen asobjective, the finite amplitude that we actually achieve for the orderwill be bound entirely by the constraints of the problem, which meansthat the number of constraints that are made binding (“activatedconstraints”) must equal the number of transmission variables that canbe adjusted. We will now show how to ensure that most of these bindingconstraints are constraints on maximum and minimum rectangletransmittance, thus forcing most rectangles to one of the bounding (andsupported) transmission limits when these constraints are activated.

FIG. 12 shows a working graylevel mask design for FIG. 8 isolationpattern; this (non-manufacturable) mask will maximize diffractedintensity. A grid spacing of α=0.15 has been chosen. Note the roughsimilarity to FIG. 7. The target mask technology is binary, with allowedtransmissions of +1 and −1. Rectangles that have been successfullydriven to a transmittance of +1 are shown as full red (indicated in FIG.12 as “R”), while −1 rectangles are shown as full blue (indicated inFIG. 12 as “B”). A minority of rectangles must be adjusted tointermediate graylevel transmittances in order to achieve the wavefrontshape specified in FIG. 9; these are shown in intermediate shades. Theminority of blocks cannot be fabricated with the specified masktechnology, and don't show the full transmission discontinuity Δt=2against adjacent rectangles. However, the above working mask design isdeveloped into a successful solution during later steps of our maskdesign algorithm, see text. Even after the first step shown above, 63%of the transmission discontinuities across gridline sections have beenmade consistent with the fabrication technology, i.e., Δt across 63% ofthe sections is 0 or 2.

FIG. 13 shows one octant of the FIG. 12 mask, along the top edge, to theright of the centerline. Rectangular gridlines are shown explicitly. Theoptical period of this DRAM level is quite large in λ/NA units (seetext), corresponding in area to two circuit features; however patternsymmetry strongly limits the independently adjustable area.Corresponding behavior is seen in the pupil domain, where half theorders have zero amplitude, and the others exhibit fourfold symmetry.(FIG. 9 lists the independent set.)

The directional separation between diffraction orders will be λ/P_(x) orλ/P_(y). The number M of collected orders will then satisfy(approximately):

$\begin{matrix}{{M \cong \frac{\pi \; N\; {A^{2}\left( {1 + \sigma_{M}} \right)}^{2}P_{x}P_{y}}{\lambda^{2}}},} & \lbrack 4\rbrack\end{matrix}$

where τ_(M) is the maximum relative obliquity of the illumination withinthe pupil (“maximum pupil fill”). The number of transmission variables Lwill equal the number of mask rectangles formed by the gridlines. Sincethe average grid separation is αλ/NA, we have

$\begin{matrix}{L \cong \frac{N\; A^{2}P_{x}P_{y}}{\alpha^{2}\lambda^{2}}} & \lbrack 5\rbrack\end{matrix}$

When we maximize intensity we require that all orders stand in aspecified ratio against the reference order; this requirement representsM−1 constraints. We have seen that a total of L constraints must beactivated (i.e. made binding) by the maximization. All M−1 amplitudeconstraints must be binding when the correct wavefront shape isachieved; the remaining L−M+1 binding constraints must come from theremaining constraints of the problem, namely those requiring that therectangle transmittances fall within the range supported by the masktechnology. Thus, the fraction f of the rectangles that is successfullydriven to supported transmission limits is

$\begin{matrix}{f = {{\frac{L - M + 1}{L} \cong {1 - \frac{M}{L}}} = {1 - {\pi \; {\alpha^{2}\left( {1 + \sigma_{M}} \right)}^{2}}}}} & \lbrack 6\rbrack\end{matrix}$

according to eqs. [4] and [5]. (One qualification here; for a Levensontrinary mask we would add a constraint that the sum of the absolutevalues of the transmittances be limited, in order to force a separationbetween some +1 and −1 regions. [Such a constraint can be linearized.]The number of binding transmission constraints is changed slightly inthis case, but the basic point that most rectangles achieve supportedtransmissions remains valid.) If α is chosen according to the criteriadescribed above, f will be greater than 0.5, so that most of therectangle transmittances will take on supported values. Eq. [6]indicates that smaller values of o should preferably be chosen whenσ_(M) is large, though the appropriate shrinkage is actually less thanquadratic.

Thus, the initial intensity maximization step generates a mask whosetransmittance over the majority of its area is supported by the nominalmask technology. (Note that at this point some aspects of the patternmay be invalid for other reasons; see below. Transmissiondiscontinuities due to EMF are likewise not yet corrected.) This can beseen for our sample problem in FIG. 12.

Note that the eq. [6] ratio can be maintained even if symmetry reducesthe number of independent collected orders, since under conditions ofsymmetry one would reduce the mask field to include only the portionthat can be set independently under the symmetry. For example, in ourDRAM example one octant of the (large) optical period constitutes such aportion (with the bilateral symmetries about x and y and the staggeredsymmetry each contributing a factor of 2); see FIG. 13.

One other point should be mentioned here—It may not be possible togenerate a particular arbitrary wavefront unless the mask technologysupports some form of phase-shifting (e.g. Levenson, attenuated-PSM[attenuated “phase-shift-mask”], chromeless). For example, a specifieddiffraction pattern cannot be generated with a classicalnon-phase-shifting chrome mask unless the zero order is brighter thanall the other orders. The linear programming step in our procedure wouldflag this when presented with such a case (i.e. it would indicate thatno solution can meet the constraints). However, whenever the wavefrontis feasible for non-phaseshifting masks our procedure will be able tocarry out the design.

The rectangles in the FIG. 12 solution that achieve the binding +1transmission are all contiguous, likewise for the rectangles that havereached the −1 transmission limit. The reason for this can be understoodby considering a hypothetical contour plot over the cell area of the“intensity” penalty involved in switching the polarity of a single boundrectangle at each mask position within the cell (in each case optimallyre-adjusting the other rectangle transmittances to maintain wavefrontshape). We expect this penalty to typically be largest at local maxima(in absolute value) of the bandlimited mask transmission as filtered bythe lens (including the NAs of both the illumination and collectionpupils). Alternatively, we can discuss the behavior of the algorithm bysupposing that we have prepared a plot of the penalty imposed on theorder amplitude that has been chosen as the objective function duringintensity maximization, expressed per unit of impulsive amplitude changeat any point (the penalty being negative when the amplitude change isnegative, making the penalty a signed quantity). The positions at whichour intensity maximization routine will place those rectangles whosetransmission must be detuned from a binding limit will all lie on thecontour of minimum sensitivity in such a plot, since this allowswavefront shape to be achieved with minimum impact on intensity. Thescale at which this sensitivity function varies is determined by thescale lengths at which the collected diffraction orders oscillate acrossthe mask plane, and the most rapidly varying of these essentiallydetermines the lens resolution. Thus, each “lobe” of the sensitivityfunction that lies within a single loop of the minimum sensitivitycontour will have a size of ˜½ the lens resolution (to order ofmagnitude). [The individual features of the known FIG. 7 solution can bethought of as the pedestals of these lobes.] Rectangles that lie mostlywithin the interior of a lobe will have a transmission at one of theextreme supported limits, while the M−1 rectangles with unsupportedtransmissions will be distributed around the boundaries of the lobes.The rectangles clustered within the interior of a lobe will show nodiscontinuity across the gridlines between them, whereas largerdiscontinuities will be present at the boundaries of the lobes, in manycases reaching one of the discontinuities allowed in the masktechnology.

We now consider the next (key) step of the method. Because of ourinitial choice of α, the M−1 rectangles whose transmission remainsunsupported will be subresolution in size. This means that if we splitone of these rectangles into two rectangles whose average transmissionmatches that of the parent, the diffracted wavefront will be almostunchanged in the collected orders. Recall that during the initialintensity maximization the transmission of the parent rectangle (whichwe will denote t₀) has been forced to lie between two supportedtransmissions, for example t_(max) and t_(min); this means that we canset the transmission of the two daughter rectangles to t_(max) andt_(min) while providing the desired average transmission of t₀ if wedivide the parent rectangle into areas having ratio(t_(max)−t₀)/(t₀−t_(min)). Such an area division represents one strategyfor (almost) completing the determination of a feasible mask—Everyremaining rectangle of intermediate transmittance could simply besubdivided in this way. The result would come quite close to providingthe correct diffraction pattern. However, such a solution would containa larger number of fragments than necessary, and in some cases the smallfragments could have a completely unsupportable topology, i.e. as verysmall gaps separating two features of the same polarity.

To prevent such problems we choose the more robust approach of evenlybisecting the blocks of unsupported transmission that lie within thesame row or column by inserting a single new gridline through the row orcolumn in question (thus adding this gridline to the variable grid). Thegridlines are inserted to evenly split all rectangles within the row orcolumn. The newly created rectangles do not have their transmissionsimmediately reset at this point; instead, the rectangle transmissionsfor the entire revised grid are recalculated by re-running the intensitymaximization algorithm.

FIG. 14 shows the result of re-maximizing intensity in the FIG. 12 maskafter bisecting those rows and columns which contain many rectangleshaving unsupported transmission values. In this case one row and onecolumn in each octant have been bisected. Note that, compared to theFIG. 12 mask, a much greater fraction of the mask now has transmissionvalues that are supported in the fabrication technology, and that thetransmission of most other rectangles comes somewhat closer to supportedvalues.

To avoid undue fragmentation of the mask patterns it is preferable thatone only inserts new bisecting gridlines down a carefully selectedminority of the rows and columns of the array, namely those rows andcolumns which contain the largest number of rectangles havingunsupported transmission values. Our even bisection of the row or columnmakes the smallest of the resulting daughter rectangles as large aspossible; nonetheless, if the algorithm has been run for more than oneiteration (see below), it is possible that certain rows or columns wouldbe left with unacceptably small daughter rectangles after bisection.This would usually depend on whether or not the row or column containsvery narrow rectangles whose transmission has already been set to asupported value, and which lie between rectangles of opposite polarity.Such rows and columns should not be considered for bisection. On theother hand, rows containing rectangles with “touching corners” (e.g.rectangles touched at a corner by a rectangle of the same polarity butadjacent along the two gridlines that cross at the corner to rectangleshaving another polarity) should be given extra priority for bisection.Otherwise, rows and columns should be considered for bisection on thebasis of whether or not they contain more rectangles with unsupportedtransmissions than other rows and columns. As a rough rule of thumb, thetotal number of graylevel rectangles in the rows and columns selectedfor bisection by the newly inserted gridlines (allowing rectangles to becounted twice) should preferably be between about M/2 or M; this can beused as a criterion for choosing the number of rows and columns to bebisected. Our method is most efficient when rectangles havingunsupported transmissions are distributed in such a way that a smallnumber of inserted gridlines can bisect a large number of rectangles.Fortunately, such distributions tend naturally to occur because thepatterns involved are non-random, causing the rectangles in question tobe non-uniformly distributed, i.e. clustered to some degree into a fewrows and columns. To help ensure such an outcome one would want to avoidchoices for α that cause the number of graylevel rectangles to be farlower than the number of available rows or columns. According to eqs.[4] and [5], taking into account the strong clustering of unsupportedrectangles that is found in practice, we would prefer to avoidconditions where

$\begin{matrix}{\alpha {\operatorname{<<}\frac{{\lambda/N}\; A}{\pi \sqrt{P_{x}P_{y}}\left( {1 + \sigma_{M}} \right)^{2}}}} & \lbrack 7\rbrack\end{matrix}$

Even with small fields ˜λ/NA, such a situation could only be the resultof choosing an overly fine initial fragmentation (and for such smallfields and fine fragmentations it would not even be particularlyimportant that the algorithm be efficient).

After bisection, the intensity maximization algorithm is run again,which causes many of the newly formed rectangles to be set to allowedtransmission values. In addition, the M−1 rectangles that need to beadjusted away from these values in order to meet wavefront constraintswill now tend to have transmissions that at least come closer tosupported values, because during the second intensity maximization thereare more adjustments available to the algorithm in the specific regionswhere shape-adjusting transmission changes are most effective. Moreover,the rectangles that are adjusted away from allowed values will oftenhave half the area as in the previous maximization.

The result of the re-maximization for our sample problem is shown inFIG. 14.

At this point it is sometimes possible to complete the solution byrounding all rectangle transmissions to the nearest supported value, andthen adjusting all grid separations (gridline positions) using a localoptimizer. Since the gridlines span the full length or width of theindependent field, the shape-handling constraints for this optimizationcan be quite simple. For example, two gridlines that contain betweenthem a feature of one polarity which separates rectangles of anotherpolarity should not be brought closer together than the minimumlinewidth or spacewidth permitted by mask groundrules, though someviolation may be permitted prior to the step of edge optimization.Rectangles that are attached to another feature along a single narrowedge (i.e. rectangles that are essentially serifs) cannot have too longan aspect ratio across the other dimension; such situations imply aconstraint on the separation of the associated gridlines.

However, there is one kind of shape constraint that cannot be dealt withusing simple gridline constraints, namely a requirement that the masknot contain rectangles with “touching corners” (see above).

To eliminate touching corners we need to modify the rounding step. Wecontinue to round-off all rectangles whose transmissions are close tosupported values; for example, all rectangles whose deviation from thenearest supported value is less than half the deviation from the nextclosest supported value. Touching corners involving any of the roundedrectangles are removed by rounding one of the rectangles to a differentallowed value (revising whichever one produces the least increase inwavefront error). If this preliminary rounding of graylevel rectangleshaving near-valid transmissions leaves more than, for example, 10unrounded rectangles, the rounding criterion is broadened to reduce thenumber of unrounded rectangles.

Next, all combinations of polarity choices for these unroundedrectangles are considered. Each combination can be evaluated veryrapidly, so that for e.g. a binary mask 2̂10=1024 combinations are easilyhandled. Combinations that include rectangles which touch at corners areexcluded from the evaluated set. Rounding combinations that providerelatively poor matching to the wavefront shape are excluded; forexample the least accurate 50% of the possible combinations. From amongthe remaining combinations one can select the particular combinationthat has the fewest right-angle polygon corners; this option is aimed atachieving the simplest possible mask shapes. The assessment criterionhere is essentially the total number of shape corners, considering thatinternal gridline intersections within the larger shapes formed by therectangles should be ignored, along with gridline intersections that liealong the straight edges of the larger shapes; only true corners arecounted.

Once the optimal rounding combination is chosen, the algorithm optimizesthe gridline separations as described above. This gridline optimizationstep may fully solve the problem; in such cases it can completely cancelthe effect of rounding error and thus achieve the desired wavefrontusing a manufacturable mask pattern. That is the case with out testproblem, as shown in FIG. 15.

FIG. 15 shows the final chromeless mask solution obtained by applyingrounding steps and optimization of gridline placement to the FIG. 14working solution, as described in the text. Amplitudes prescribed inFIG. 9 are achieved exactly.

Other methods for rounding can be adopted, such as error diffusion, inwhich any error (area-weighted) that arises when a rectangle'stransmission is rounded gets subdivided and distributed among adjacentrectangles that have not yet been rounded. Since the rectangles aresubresolution this allows rectangles to partially compensate eachother's rounding error.

When an exact solution is not achieved by gridline optimization, ourprocedure follows one of two paths. If the maximum error in any of theorders is less than about 0.3 (on a scale where the average orderamplitude is about 1), the algorithm attempts an exact solution byoptimizing each edge location independently, no longer requiring theedges to lie on common gridlines. This optimization can be attemptedwith constraints that enforce manufacturability. The remaining error inthe wavefront is now so small that we have a reasonable chance of rapidconvergence using only very small movements of the edges off the formergridlines.

If the residual wavefront error is larger than about 0.3, or if theoptimization against edges does not converge (or does not convergerapidly), then the algorithm simply runs through another iteration ofthe above variable grid steps (i.e. intensity maximization, bisection,re-maximization, rounding, and gridline optimization); see the FIG. 10flowchart. Another alternative, though not the preferred embodiment, isto optimally divide each unrounded rectangle arising during the seconditeration with an independent edge, as opposed to evenly bisecting allsuch rectangles in particular rows or columns using field-spanninggridlines. As noted earlier, such an approach could lead to excessivefragmentation if employed during the first iteration, but is morereasonable in later iterations.

We can use our test problem to illustrate these two alternativealgorithm paths by deliberately operating the earlier stages of thealgorithm using inappropriate parameters. We will choose an overlycoarse initial gridline spacing (of α=0.26), and we will insert too fewnew bisecting gridlines, so that fewer than M/2 of the rectangles havingunsupported transmissions will be intercepted. (To be specific, we willadd only one bisecting gridline instead of two.) When the algorithm isrun in this way we obtain the working solution shown in FIG. 16.

FIG. 16 shows intermediate results for the FIG. 8 problem when thealgorithm is deliberately run with inappropriate parameters. This isdone to explore algorithm performance when convergence is slow, seetext. Algorithm has been run with overly coarse initial grid spacing,and with too few row, column bisections. Working mask is shown aftersecond intensity maximization, same stage as FIG. 14.

Continuing through the steps of the method, we next round thetransmissions of the FIG. 16 rectangles using the combinatorialprocedure described above, and then optimize the gridline positions;this provides the intermediate solution shown in FIG. 17.

FIG. 17 shows the result after one stage of main loops when running theFIG. 8 problem using inappropriate parameters (to test the algorithmwhen convergence is slow). The FIG. 16 solution has now been furtherdeveloped to the stage where gridline separations (but not corners) havebeen optimized. This immediate solution has not yet achieved thewavefront prescribed in FIG. 9; the maximum amplitude error is about0.9.

FIG. 18 shows one octant of the FIG. 17 mask, along the top edge, to theright of the centerline. Rectangular gridlines are shown.

The FIG. 17 intermediate solution fails to achieve the FIG. 9 targetamplitudes; the maximum error is about 0.9, and the Root Mean Square(RMS) error about 0.5. When the residual error from an intermediatesolution is as large as this, the algorithm of the preferred embodimentwill cycle the intermediate solution in a second pass through theearlier steps of the procedure of FIG. 10. However, we can explore therobustness of the method by instead attempting an optimization ofindividual feature edges. The solution obtained by this choice FIG. 19actually succeeds in achieving the prescribed amplitudes. However, theedge positions have had to shift very substantially during theoptimization, and have reached a configuration that would be somewhatdifficult to fabricate, because the “vertical bars” on the “T” featureshave a long aspect ratio yet are quite narrow.

Given the large errors in the FIG. 17 solution, the algorithm shouldproperly have taken another iteration through the maximization andbisection steps, rather than carrying out an optimization of the featureedges. FIG. 20 shows the result of this 2nd iteration, which remainsnon-standard due to earlier steps in the example. Further, to maintainconsistency with FIG. 10, the iteration has again been carried out usingan inappropriately small level of bisection, i.e. as before only one rowor column has been bisected. Next, the algorithm applies rounding andoptimization of gridlines and edges to the FIG. 20 solution, achievingthe successful result shown in FIG. 21. Even though the procedure hasbeen run with non-optimal parameters, a reasonable solution has beenobtained.

Referring now to FIG. 19, the poor convergence of the FIG. 17 solutionwould normally call for a second stage of intensity maximization andgrid bisection, but to explore algorithm behavior we have attempted anoptimization of individual feature edges, as shown here. This solutionsucceeds in achieving the FIG. 9 amplitudes; however the feature edgeshave moved quite far from their FIG. 17 initial positions, into aconfiguration that would be somewhat difficult to fabricate, due to thenarrow (31 nm) vertical bars.

FIG. 20 shows the working mask solution obtained after cycling the FIG.18 gridlines through another iteration of intensity maximization,bisection, and re-maximization. This second iteration is the appropriatepath for the algorithm, given the poor performance of the FIG. 17gridline optimization. However, during the steps of this seconditeration we continued, for testing purposes, to employ the sameinappropriate algorithm parameters as were used during the firstiteration, which produced FIG. 16.

FIG. 21 shows the solution obtained from the FIG. 20 intermediate resultafter rounding and optimization of gridlines and edges. The FIG. 9wavefront prescription is achieved using reasonable features, eventhough the algorithm has been operated in a non-optimal way.

We conclude the discussion of optimization of the thin mask edgediscontinuities by showing a mask example FIG. 22 that is calculated fora larger field, namely an SRAM (Static Random Access Memory) CA level,with optical pitch 2.16 μm×0.8 μm, to be exposed at λ=193 nm andNA=0.75. The mask is atten-PSM with 10% background transmission. 41orders are collected by the lens. For comparison the figure shows theFIG. 15 DRAM solution as a black-and-white insert, scaled to the correctrelative size in λ/NA units.

In FIG. 22, the upper color plot shows another example of the invention,namely a relatively large-field mask for a (Static Random Access Memory)SRAM contact hole level, see text, designed by our method for 10%atten-PSM. The black-and-white lower insert reproduces the FIG. 15 DRAMsolution at a scale commensurate with the large-field cell when measuredin dimensionless λ/NA units.

FIG. 23 shows a brick pattern used to illustrate the process of EMFcorrection according to an embodiment of the present method.

EMF effects are not handled under the procedure as described to thispoint, since we have thus far assumed that the discontinuities at thetopographic edges separating features of different polarity are equal tothe difference in the nominal transmissions of the two regions involved(with these nominal transmissions being supported by the mask technologywhen the above procedure completes). This impact from lack of EMFcorrection is illustrated in FIG. 24.

FIG. 24 shows EMF and TMA images of the FIG. 23 mask. The “EMF”-labeledcurves are calculated using an accurate Maxwell simulator. Maxwellsimulator refers here to software of a well-known kind that numericallyor analytically solves Maxwell's equations in order to determine themost accurate possible electromagnetic field that is produced when aninput optical field, such as a plane wave, interacts with a structure.Despite being optimally refocused and renormalized, they differappreciably from the target “TMA”-labeled curves. (The thick curves arevertical cutlines, the thin curves horizontal. Solid curves plot theintensity across the centerlines of the image, while the dashed curvesplot the intensity along the period boundaries.) Here λ=193 nm andNA=1.2.

To introduce control of EMF we can employ two different approaches: Inthe first approach we adjust the target diffraction orders in such a wayas to render the feature discontinuities in as closely equivalent a formas is possible to those of the nominal thin mask, under the limitationthat the mask topography remain fixed, and that only the 2D shapes ofthe patterns are subject to adjustment. This is effective in controllingthe in-phase part of the EMF-induced edge discontinuity. Our secondapproach employs phase shifters (of new polarity) on the mask in orderto suppress the in-quadrature part of the EMF-induced edgediscontinuity.

We now consider the first of these approaches, in which we find shapesthat (when rendered in a physically realistic topographic mask) willhave edge discontinuities that are brought as closely as possible tothose of the nominal thin mask target. (Here the nominal thin masktarget is one which, per the procedure described above, successfullyprovides the desired wavefront, but only under the simplified assumptionof a TMA model.) To carry out this procedure we need a metric forjudging the closeness of the real-valued TMA discontinuities of thenominal mask to the complex discontinuities that the topographic maskwill produce.

Treated collectively, the set of feature discontinuities in thetopographic mask will be brought as closely as possible to those of thethin mask target when the images produced by the two masks have asclosely matched intensity as possible. Thus, we could in principledefine the EMF-corrected mask as being the output of a slightlycumbersome optimization procedure, in which (beginning with the TMA masksolution) we iteratively calculate the EMF image, and make adjustmentsin the mask shapes, in such a way as to minimize the deviation of theimage from the target image. Such an optimization problem becomescomputationally quite tractable if approximate methods like boundarylayers are employed, but the required computation of a partiallycoherent image at each iteration is more cumbersome than necessary.

FIG. 25 shows an EMF-corrected version of the FIG. 23 mask.

However, we now show that by using a frequency domain method we canobtain the EMF-corrected mask shapes in a simpler way. Using a 1Dcoherent example to simplify the notation, we can write the vector imageof a periodic object (period P) as

$\begin{matrix}{{I(x)} = {\sum\limits_{m}^{\;}{\sum\limits_{n}^{\;}{\left\lfloor \begin{matrix}{{F_{m,n}^{X - {Pol}}M_{n}^{X - {Pol}}M_{m}^{X - {Pol}^{*}}} +} \\{F_{m,n}^{Y - {Pol}}M_{n}^{Y - {Pol}}M_{m}^{Y - {Pol}^{*}}}\end{matrix} \right\rfloor ^{2\; \pi \; \; {{x{({m - n})}}/P}}}}}} & \lbrack 8\rbrack\end{matrix}$

assuming unpolarized illumination as an example. Here M designates thepupil-plane amplitude of the electric field vectors, and superscriptsX−Pol and Y−Pol indicate which of the two independent polarizationcomponents of the illumination is being considered. The F_(m,n)coefficients include obliquity factors, polarization aberrations fromthe lens and resist stack, and a dot product between the image-planeunit vectors of the interfering orders. Defining

G_(m,n) ^(X−Pol)≡F_(m,n) ^(X−Pol)M_(n) ^(X−Pol)M_(m) ^(X−Pol)*   [9]

and similarly for Y−Pol, we can write the difference between two imagesI(x) and I₀(x) as

$\begin{matrix}{{{{I(x)} - {I_{0}(x)}} = {\sum\limits_{m}^{\;}{\sum\limits_{n}^{\;}{D_{m,n}^{2\; \pi \; \; {{x{({m - n})}}/P}}}}}}{where}} & \lbrack 10\rbrack \\{D_{m,n} \equiv {G_{m,n}^{X - {Pol}} - {{}_{\;}^{}{}_{m,n}^{X - {Pol}}} + G_{m,n}^{Y - {Pol}} - {{{}_{\;}^{}{}_{m,n}^{Y - {Pol}}}.}}} & \lbrack 11\rbrack\end{matrix}$

Using the Fourier transform of a delta-function, we find after somealgebra that

$\begin{matrix}{{\int\; {{x}{{{I(x)} - {I_{0}(x)}}}^{2}}} = {\sum\limits_{N}^{\;}{{\sum\limits_{\underset{L + {N\mspace{14mu} {even}}}{L}}^{\;}D_{\frac{L + N}{2},\frac{L - N}{2}}}}^{2}}} & \lbrack 12\rbrack\end{matrix}$

Though the focus and source dependence has been suppressed for brevity,it is straightforward to average eq. [12] through focus, and likewiseeq. [9] can be averaged over the source. The M coefficients for thetopographic mask need not be assumed independent of source direction;however this independence obtains by definition for a TMA mask. Fastsimulations of topographic masks usually reduce them to approximate TMAequivalents (e.g. using boundary layers), but one can optionallycalculate the G coefficients for the topographic mask using a rigorousMaxwell solver. Here G represents the strength of the intensityoscillation that is produced in the image when a projection lens causestwo particular waves from a mask to interfere. Also, “Maxwell solver” isa synonym for “Maxwell simulator”.

To apply eq. [12], we first set I₀ equal to the image produced by thetarget wavefront. The TMA design provided by the FIG. 10 procedure willproduce this I₀ image under the TMA approximation. In one embodiment ofthe present method, these same shapes are next rendered as a topographicmask. A local optimizer then adjusts the shapes of the topographic maskfeatures to minimize eq. [12] while maintaining mask manufacturabilityusing nonlinear constraints on the shape variables. In cases where allpatterns of interest are being handled simultaneously by the optimizer,one should preferably treat focus as an additional variable during thisoptimization.

When the EMF mask is simulated using an accurate Maxwell solver, it canbe efficient to stage the (usually time-intensive) EMF calculation. Oneway to do this is to use scalar Fourier offsets as optimizationvariables (one per collected diffraction order), and then to use eq.[12] to calculate the specific values of these offsets which, when addedto the diffraction orders collected from the topographic mask, cause theimage to resemble as closely as possible the I₀ image from the targetwavefront. No new EMF calculations are needed to evaluate eq. [12]during this calculation, making its minimization quite rapid. Morespecifically, in this embodiment eq. [12] takes the form of a 4th orderpolynomial in the offset variables, and because EMF effects typicallyamount to a modest perturbation on the TMA solution, eq. [12] can berapidly minimized by iterating towards the particular local minimum ofthis polynomial that is closest to the origin (in the space of theoffset variables).

Once the optimum offsets are found, we next apply them to the wavefronttargets used in the FIG. 10 wavefront engineering procedure. Since EMFeffects tend to be perturbational, it is appropriate to only rerun thefinal “Optimize Edges” step of this procedure. The resulting revisedmask patterns will generally not be greatly different from those of theTMA solution, and as a result the EMF effects generated by the revisedpatterns (after realistic topography is considered) will not be greatlydifferent from those generated by the original solution (which neglectedrealistic topography). Of course, if the change in EMF were so small asto be literally negligible, our revised patterns would constitute thedesired EMF-corrected solution to the wavefront engineering problem,i.e. the revised patterns would provide optimal edge discontinuities.

As shown in FIG. 26, the blue (indicated as “B” in FIG. 26) curve plotsthe image of the FIG. 25 mask as calculated using a rigorous Maxwellsolver. This image matches almost exactly the target image shown in red(indicated as “R” in FIG. 26), which is produced by the FIG. 20 maskunder a TMA model. A very accurate correction is achieved after only asingle iteration of the procedure described in the text.

However, in many cases the change in EMF effects, though small, is largeenough to matter, and in such cases it is desirable to iterate the aboveprocedure, i.e. to determine optimal values of a new set of offsetvariables that re-minimize eq. [12], and then to rerun the final stageof the FIG. 10 procedure in an additional iteration, yielding re-revisedmask shapes that successfully provide the new offsets. This process canbe iterated until the changes in EMF induced by the latest round ofshape revisions are negligible.

In cases where all patterns of interest are simultaneously EMF-correctedusing this procedure, it may be desirable to optimally readjust thefocus that is assumed for the topographic versions of the patterns(relative to the focus at which I₀ is calculated). This maybe done byapplying a least squares fit to the phase difference of each orderrelative to those of the real-valued TMA mask. If the time origin forthe fields is not maintained consistently between the TMA and EMFcalculations, one should include a constant (piston) term in the phasefit.

The above method can also account for other mask nonidealities besidesEMF effects, such as corner rounding and dimensional distortions in thepolygons that the maskwriter actually fabricates in the mask (whennominally Manhattan design shapes are specified). As with EMFcorrection, it is necessary that one be able to calculate or estimatethe nonidealities of interest.

When used to correct EMF, the procedure just described can provide thebest possible adjustment of the (2D) positions of the feature edges onthe mask, in order to minimize the deviation of the mask image from theimage produced by the desired wavefront. In most cases the match willnot be perfect. EMF effects distort the transmission of the mask in thevicinity of feature edges, and the distorted transmission is typically acomplex-valued quantity (even though the TMA transmission is pure real).At distance scales that the projection lens can resolve (which are thescales that matter as far as the projected image is concerned), thein-phase component of the edge discontinuities of the repositioned edgesessentially matches those of the TMA mask designed by the FIG. 10procedure (if averaged through focus). However, the quadrature componentof the EMF-induced transmission distortion is not corrected (oraffected) by the edge-position adjustment.

As noted above, the present method includes a second method foreffecting EMF correction, namely to adjust the topographic structure ofeach 3D edge itself in order to eliminate the in-quadrature component ofthe EMF-induced distortion. Note that while it might appear morestraightforward to contemplate complete removal of all EMF-induceddistortion, such a brute-force correction would appear to be extremelycomplicated and difficult; indeed, no specific structure is know toeffect such an EMF suppression. However, the present method is able toimplement two significant relaxations to this brute-force suppressionwhile still achieving substantial correction of the image. First, ourmethod only requires a correction which causes the edge-structure tomatch that of the TMA edge when viewed under the very limited resolutionof the projection lens (which might roughly be ±0.3 μm at the maskconjugate, i.e. the lens resolution patch encompasses a considerableregion in the neighborhood of the physical edge). Second, this aspect ofthe present method only needs to correct the in-quadrature component ofthe EMF-induced distortion; we use the above shape-based adjustmentmethod to correct the in-phase component.

Because it can exploit these two relaxations, the present method is ableto effect the correction using standard fabrication processes. Consider,for example, a mask that is fabricated by etching polygonal aperturesinto a film that covers an SiO2 substrate, so that patterns are formedwhere the opened film apertures expose the transparent SiO2 substrate.By using a second patterning step to leave a narrow pedestal region in(or to etch a narrow trench region into) the SiO2 at a position veryclose to an existing vertical edge, we are able to give the near-fieldtransmission a value that is phase-shifted, i.e. that has a transmissionwith non-zero in-quadrature part. If the narrow feature does not extendfar enough from the aperture edge for the lens to resolve it, the edgewill effectively have the same null in-quadrature component in itstransmission discontinuity as would a nominal TMA edge. In effect, thenarrow region is functioning as an anti-boundary-layer (anti-BL), thoughas noted it need only cancel the EMF-induced boundary layer in itsquadrature component, and only in those spatial frequencies which thelens can resolve.

FIG. 27 shows basic anti-BL configurations.

Of course, there is no rigid dividing line between the dimensions that alens can resolve and those that it cannot, and typically thetransmission correction will be imperfect since the new narrow regionextends a finite distance away from the edge. However, as will be shownin examples below, anti-boundary-layers (anti-BLs) of width 80 nm andmore can effect a very substantial correction of EMF effects.

Moreover, perfect correction in each polarization can (within theHopkins approximation) in principle be attained if the mask contains twonew kinds of features whose different nominal phase shifts providenon-zero in-quadrature components of opposite sign. This conclusion isin no way invalidated by the fact that the variously-phased maskstructures will interact with each other in a very complicated manner,nor by the fact that the boundaries between regions of each kindrepresent distinct topographic discontinuities. Of course, thesecomplexities increase the computational burden, but fundamentally thecorrection process is simply that of the FIG. 10. TMA wavefrontengineering procedure, except carried out for both the in-phase andin-quadrature components of the diffracted fields, and with the provisothat the adjustments be “over-shot” in such a way that the follow-on EMFeffects which they themselves introduce are cancelled along with thosefrom the primary aperture edges. In fact, in the absence of incidenceangle dependencies, it is possible to carry out this correction usingonly three phase polarities in the mask, as long as the ratio of thephase shift from two of the regions to that of the third providesin-quadrature components of opposite sign.

However, as with the above shape-correction method, it is useful toexploit the fact that EMF effects are typically perturbational incharacter. This causes the etch depth or pedestal height required in theanti-BL to be small, meaning that the new topographic discontinuitiesthat the anti-BL itself will introduce are only 2nd order (though ifconsidered excessive they can optionally be corrected iteratively). Aswith the conventional boundary layers used in simulation, it is oftenacceptable to merely deploy a uniform anti-BL along every topographicedge. FIG. 28 shows that with chromeless masks (T=±1) a single extrapatterning step suffices to create anti-BL's that are centered on thenominal edge.

Moreover, FIG. 28 shows that with a chromeless (CPL) mask a singleadditional patterning step can create anti-BL's that are centered on thenominal edges. (The in-quadrature parts of the anti-BLs share a commonsign if only one extra patterning is used.) Gray shading is used toindicate areas that are not etched in each step.

An important additional consideration here is the so-called isofield. Ina separate disclosure, the inventors have shown that even when theillumination is unpolarized, we can use a single iso-edgefield toaccount for the EMF effects induced by topography. Specifically, we usea coherent weighted sum of the edge-fields for TE and TM orientations,even though the illumination is unpolarized. This approach can bejustified mathematically as long as EMF effects are small. An importantimplication is that a correction approach derived for polarizedillumination can be made to work in the unpolarized case as well, ifapplied to the EMF-induced iso-field.

Since the anti-BL width is too small for the lens to resolve, it is onlyits net contributed quadrature component that is important for EMFcorrection. It is therefore usually not critical to choose preciselyboth the width and the height (or depth) of the anti-BL, but only theircombination.

While the depth of the anti-BL may be set at a fixed quantity, the widthof the anti-BL maybe more easily adjusted (if the anti-BL is created bya patterning step rather than a self-aligned process). This allows us topartially compensate such higher order effects as feature-to-featureinteraction, the so-called “non-Hopkins” dependence of transmission onincidence angle, and changes in polarization introduced by roundedaperture corners or compound illumination angles. More complexstructural changes may also be employed towards this end, such asvariations in the thin film design of the mask blank, or deposition ofadditional films along the sidewalls, or changes in the sidewallprofile. It is also possible to bring all transmission discontinuitiesinto conformance with the nominal value supported by the mask technologyby adjustment of both the phase shift of the mask blank film stack, andthe density of patterns on the mask, including non-printing patterns inmost cases. As discussed above, the transmission discontinuities willachieve the desired conformance if the bandlimited rendition of the mask(as filtered by the lens) achieves substantially the same shape as theinverse Fourier transform of the specified wavefront. This means, forexample that in the special case where the features are uniformly spacedwith a separation that happens to equal the width of two anti-BLs, theentire space between each feature pair would in fact take the form of ananti-BL. In such a mask the nominal background polarity would thereforebe absent, meaning that the anti-BL polarity is essentially serving asthe effective background polarity. Such masks are therefore essentiallyequivalent to masks whose nominal film stack transmission is detunedfrom a phase thickness of 180° (after suitably readjusting the fixedfeature separation to account for the associated film stack topographychange), but whose actual edge discontinuities achieve conformance withthe desired 180° transition due to EMF effects.

Of course, the above scenario assumes that the features have a suitableuniform density. But since the transmission discontinuities need onlyachieve their target values after filtering by the bandlimitedresolution of the lens (including the illumination NA), we can alsoinclude sub-resolution features to help achieve the necessary density.In many cases these assist-like features need to be accounted for duringthe initial design of the wavefront, but it is known that smallsub-resolution features can benefit image quality.

We refer to this approach of adjusting both the background phase shiftand the density and positions of deployed unresolved features as“cheese-and-fill boundary layers”. Here we have borrowed the term“cheese-and-fill” from an unrelated technique in which electricallyinert features are included in patterns for the purposes of achieving adesirable density uniformity during film etch processes. It should benoted that in the limit of large features the “non-Hopkins” dependenceof transmission on illumination angle can be approximated by the angulardependence of the mask blank film stack. Suppose, for example, that achromeless mask in which SiO2 of index n has been etched to a depth thatprovides 180° phase shift for incidence angle θ₀, is instead illuminatedat a different angle θ. We can estimate that the phase shift will begiven by

$\begin{matrix}{{Phase} = {{180{^\circ}\frac{{n\; \cos \; \theta^{\prime}} - {\cos \; \theta}}{{n\; \cos \; \theta_{0}^{\prime}} - {\cos \; \theta_{0}}}} \cong {180{{^\circ}\left\lbrack {1 + \frac{\theta^{2} - \theta_{0}^{2}}{2\; n}} \right\rbrack}}}} & \lbrack 13\rbrack\end{matrix}$

where the primes indicate propagation angles inside the SiO2 as given bySnell's law. Note that this expression only applies to extended areas,and does not consider incidence angle dependencies or shadowingasymmetries at edges.

FIG. 29 shows correction of the EMF-induced feature-dependent focalshifts seen in FIG. 5 using anti-boundary-layers.

FIG. 30 shows correction of the EMF-induced focal shifts seen in FIG. 6using a negative anti-boundary-layer. At a fixed focal plane, theanti-BL produces a behavior of the mask diffraction that closely matchesthat of the TMA model.

FIGS. 31A-31B show anti-BL's to correct focus variations in printing 50nm lines at different pitches. In accordance with the simplest BL modelthe same anti-BL has been used at all pitches, which leads to inferiorcorrection at the smallest pitches when the illumination contains asignificant TE component.

FIGS. 32A-32D show anti-BL correction of a chromeless phase-shift mask(a so-called CPL mask), for a test problem. Printed CDs are shown for180 nm pitch gratings, as the duty cycle and focus are varied. The leftside shows results for the known mask, whose images suffer from strongfocal asymmetries. The right side shows that a simple pitch-independentanti-BL provides distortion-free printing through focus. Theillumination is unpolarized.

Moreover, FIG. 29 and FIG. 30 show examples of anti-BL masks thatcorrect the focal shift distortions seen in FIG. 5 and As shown in FIGS.3A-3B, respectively. FIGS. 31A-31B show another example of anti-BLcorrection. FIGS. 32A-32D show anti-BL correction of a chromeless mask.

The following is an Appendix of Orientation Independent BoundaryTopography Correction of Electromagnetic Effects in Photomasks.

APPENDIX A A.1 AIMS Asymmetry Factor Measurements Evidence ofElectromagnetic Phase Errors Cause

FIG. 33 shows a sketch of the aerial image across several focal planesproduced by a simple phase shifting grating of 350 nm pitch (as measuredon the wafer plane). When the ratio between the grating space and lineis equal to the amplitude transmission, then the TMA model of the maskpredicts a two beam diffraction pattern where the direct beamdiffraction order turns to zero. A deviation from this space to lineratio or perturbations like EMF variations induced by the masktopography introduces a non-negligible amount of zero-th diffractedorder and, in consequence a focus-dependent asymmetry will produce in anominally two-beam interference image. Specifically, the asymmetryparameter is the intensity difference that arises between adjacent peaksin the nominally sinusoidal fringe pattern (a pure harmonic when theasymmetry parameter is zero), δ normalized by the DC intensity level(i.e. the average intensity in the image), ρ.

Simulations that ignore EMF induced phase errors, that is, TMAsimulations, show symmetric plots through focus of the asymmetry factoras displayed in FIG. 34A.

Full EMF simulations and AIMS measurements [Mike Hibbs and TimothyBrunner, Proc. SPIE 06] show a distinctive linear dependence throughfocus (curve tilting) due to transmission EMF phase errors shown in FIG.34B. The acronym “AIMS” stands for Aerial Image Measurement System. Itrefers to a microscope that measures the image produced by a mask.

In FIGS. 34A-34B, the TMA fails to reproduce 3D mask effects (bothamplitude and phase), a biased TMA can account for in-phase componenterrors but fails to model in-quadrature errors responsible for theasymmetry parameter curve tilting.

A.2: Isotropic Boundary Layer

As shown in FIG. 35, the dominant non-Kirchhoff effect can be shown tobe localized near the bounding perimeter of mask apertures [J. TirapuAzpiroz and E. Yablonovitch, J. Opt. Soc. Am. A 23, 821 (2006)]. Aboundary layer can accurately model mask EMF effects due to topography(both in-phase and in-quadrature) through a strip of complex-valuedtransmission deployed during thin mask simulations in the vicinity offeature edges. The boundary layer model width and real component accountfor amplitude errors, and the boundary layer in-quadrature componentaccounts for phase errors. It can be seen that an isotropic boundarylayer model [“Efficient Isotropic Modeling Approach to IncorporateElectromagnetic Effects into Lithographic Process Simulations”.Tirapu-Azpiroz et. al. FIS8-2006-0379], formed through the weightedcoherent combination of the boundary layers due to TE and TMpolarization components of the incident illumination, can accuratelyapproximate the EMF impact due to unpolarized illumination.

A.3: EMF Correction on the Reticle

A boundary layer model can reproduce the effects of mask EMF duringlithographic simulations, but cannot correct for the degradation oncommon process window induced by the fluctuation of plane of best focusinduced by the in-quadrature component of the EMF effects.

It is shown in FIG. 36 that application of the anti-BL correction to themask edge produces a mask profile that reproduces ideal (TMA) responsethrough focus in amplitude and phase and hence can restore the focusdrift observed at tight pitches and restore CommonPW performance

A.4: AntiBL Parameters Optimization

FIG. 37 shows Measure 1: optimization through minimization of asymmetryfactor focus shift. The parameters of width and depth of the anti-BLedge correction are optimized by searching for the values that producethe a symmetric curve of asymmetry factor measurement. It can be seen inFIG. 37 that a range of depth-width pairs exist that satisfy thiscondition.

FIG. 38 shows Measure 2: optimization through minimization of Abs(imaginary (0th order/1st order)) The term “Abs.” is short for “absolutevalue”. In other words, the figure shows the absolute value of theimaginary part of the ratio of the 0th order to the 1st order. FIG. 38shows that the range of depth-width pairs that satisfy this conditionare essentially the same as those in FIG. 37 and, hence both approachescan be seen as equivalent.

FIG. 39 shows other example profiles.

FIG. 40 shows Chromeless example.

APPENDIX B Effect of Quadrature Component of the Diffracted Field onWafer Focal Plane

When the Thin Mask Approximation can be assumed to model the fieldtransmitted through a photomask with acceptable accuracy, then fornormal incidence illumination of the mask, the aerial image intensity inthe wafer plane can be expressed in the important case of 3 beaminterference imaging, as equation (B.1):

FIG. 41 shows: a) Diffraction orders diffracted by the photomask withnormal incident illumination as described by the Thin Mask Approximation(TMA), and b) Thin Mask Approximation of the mask transmittedNear-fields.

I _(image) ^(TMA) =|A ₀ ^(TMA)|²+4|A ₁ ^(TMA)|²+4A ₀ ^(TMA) A ₁^(TMA)*cos(k _(z) −k ₀)z   (B.1)

with

$k_{0} = {{\frac{2\; \pi}{\lambda}\mspace{14mu} {and}\mspace{14mu} k_{z}} = {k_{0}\left( {1 - \left( \frac{\pi}{2\; P} \right)^{2}} \right)}^{\frac{1}{2}}}$

and where, for TMA and real blank transmission (either 0 degs or 180degs), then the following relation is satisfied:

A ₀ ^(TMA) A ₁ ^(TMA) *=A ε

Hence, the plane “z” of best focus is given by the solution to equation(b.2):

$\begin{matrix}{\frac{\partial I_{image}^{T\; M\; A}}{\partial z} = {{{- 4}\; {A\left( {k_{z} - k_{0}} \right)}{\sin \left( {k_{z} - k_{0}} \right)}z} = 0}} & \left( {B{.2}} \right)\end{matrix}$

where the best focus is constant across pitch and equal to z_(BF)=zero

Similarly, the asymmetry factor is given by

$\begin{matrix}{{AsymFactor} = {8\frac{A_{0}^{T\; M\; A}A_{1}^{T\; M\; A^{*}}{\cos \left( {k_{z} - k_{0}} \right)}z}{{A_{0}^{T\; M\; A}}^{2} + {4{A_{1}^{T\; M\; A}}^{2}}}}} & \left( {B{.3}} \right)\end{matrix}$

which, when

$\frac{Space}{Line} = {\left. {{Mask}\mspace{14mu} {Transmission}}\rightarrow A_{0}^{T\; M\; A} \right. = {\left. 0\rightarrow{AsymFactor} \right. = 0}}$

hence producing a flat asymmetry factor through focus.

On the other hand, under similar circumstances but taking into accountthe full electromagnetic nature of the fields transmitted by thephotomask, the aerial image intensity at the wafer plane can beexpressed as equation (B.4)

FIG. 42 shows: a) Diffraction orders diffracted by the photomask withnormal incident illumination when considering full diffractedelectromagnetic field (EMF); b) Sketch of the photomask electromagneticnear fields.

I _(image) =|A ₀|²+4|A ₁|²+2 Re[A ₀ A ₁ *e ^(−i(k) ^(z) ^(−k) ⁰^()z)]  (B.4)

where the diffracted orders are those produced by the fullelectromagnetic interaction between the mask topography and the incidentillumination and the aerial image intensity is evaluated at x=0 forsimplicity. It is possible to express the diffracted orders produced bythe full electromagnetic interaction as the sum of a TMA term plus anEMF-induced perturbation term due to the EMF impact as follows:

A ₀ *=A ₀ ^(TMA) +ΔA ₀ ^(EMF)   (B.5a)

A ₁ *=A ₁ ^(TMA) +ΔA ₁ ^(EMF)   (B.5b)

where now the terms ΔA₀ ^(EMF)=Re(ΔA₀ ^(EMF))+ilm(ΔA₀ ^(EMF)) and ΔA₀^(EMF)=Re(ΔA₁ ^(EMF))+ilm(ΔA₁ ^(EMF)) have both in-phase andin-quadrature components, while the TMA term remains purely in-phase,that is, A₀ ^(TMA)=Re(A₀ ^(TMA)). Then the cross product of the zerothand first diffracted orders is no longer purely real and it will containboth amplitude and phase terms (or in-phase and in-quadraturecomponents) as indicated by equation B.6:

A ₀ A ₁*=(A ₀ ^(TMA) +ΔA ₀ ^(EMF))(A ₁ ^(TMA) +ΔA ₁ ^(EMF))*=B e^(iδ)  (B.6)

Hence, due to the in-quadrature component of the diffracted orders whenthe full electromagnetics are considered, an EMF-induced phasedistortion term is introduced into the aerial image expression that willproduce deviations of the best focal plane relative to the ideal z=0plane, that is, the best focal plane is not longer constant and equal tozero across pitch, instead it will depend on the feature size and pitchof the pattern being imaged (B.7):

$\begin{matrix}{z_{BF} = \frac{\delta}{k_{z} - k_{0}}} & \left( {B{.7}} \right)\end{matrix}$

and the asymmetry factor is given by equation (B.8)

$\begin{matrix}{{AsymFactor} = {{- 4}\frac{B\; {\cos \left\lbrack {\delta - {\left( {k_{z} - k_{0}} \right)z}} \right\rbrack}}{{A_{0}}^{2} + {4{A_{1}}^{2}}}}} & \left( {B{.8}} \right)\end{matrix}$

Thus the in-quadrature part of the electromagnetic fields are alsoproducing a non-symmetric plot even when A₀ ^(TMA)=A=0, since the termΔA₀ ^(EMF) and hence the term B will not likely be zero for realisticmask blanks.

The above description of the mask topography-induced focus distortionsfor normal incident illumination can be extended to oblique incidence ofthe illumination (so-called off-axis illumination) where it is knownthat these distortions or shifts of the plane of best focus across pitchcan be amplified by the oblique nature of the illumination according toequation (B.10), where

$\begin{matrix}{{\sin \; \theta_{d}} = {{\frac{\lambda}{Pitch} - {\sin \; {\theta_{inc}.z_{BF}}}} = \frac{\delta}{k_{0}\left( {{\cos \; \theta_{inc}} - {\cos \; \theta_{d}}} \right)}}} & \left( {B{.10}} \right)\end{matrix}$

Referring now to FIG. 43, there is depicted a computer 4310 having amemory 4300 storing a program of computer readable instructionsexecutable by a processor 4320 to perform actions directed to generatinga desired set of diffracted waves using features of a lithographic maskfor which a set of allowed transmissions are chosen from a set ofsupported values, the actions comprising:

-   -   creating a representation of the mask as a set of polygonal        elements,    -   defining constraints which require that the ratio of the spatial        frequencies in the representation take on the amplitude ratios        of the desired set of diffracted waves,    -   using an optimization algorithm to adjust the transmission        discontinuities at edges of the polygonal elements to        substantial equality with the discontinuity values allowed by        the set of supported transmissions while maintaining the        constraints. The computer 4310 may be a single computer (e.g.,        mainframe, personal computer, etc) or multiple distinct        computers arranged to function in a distributed-computing manner        (e.g., cloud computing). Similarly, the memory 4300 and        processor 4320 which accomplishes the above creating, defining        and adjusting may be within a single computer or the functions        detailed herein may be accomplished by a functionally organized        group of physically distinct computers each having their own        processor and memory and coupled to one another via some        communication link. Regardless of how many computers are used,        the desired set of diffracted waves (e.g., the wavefront) is        output and used to form a lithographic mask.

In one aspect of the memory 4300, the optimization algorithm comprisesiterated steps, the iterated steps comprising:

-   -   forming a 3D topographical representation from the polygonal        elements, and simulating it with a full-3D Maxwell solver to        calculate the Fourier transform of the edge discontinuities.

In another aspect of the memory 4300, the iterated steps furthercomprise:

-   -   calculating a compensating adjustment that cancels the        deviations of the Fourier transforms of the edge discontinuities        from the required spatial frequency ratios.

In a further aspect of the memory 4300 the iterated steps furthercomprise:

-   -   forming an adjusted set of Fourier orders using the compensated        edge Fourier transforms calculated in the previous step and use        them to generate with thin-mask wavefront engineering a new set        of iterated polygonal elements.

In another aspect of the memory 4300, the optimization algorithm furthercomprises

-   -   terminating the iterations when the Fourier transform of the 3D        topographical representation of the iterated polygonal elements        substantially reproduces the amplitude ratios of the desired set        of diffracted waves.

In a yet further aspect of the memory 4300, one or more transmissiondiscontinuities are driven to substantial equality with an allowed valueby:

-   -   forming the 3D topographical representation of the polygonal        elements,    -   calculating the transmission discontinuity at the edges of the        polygonal elements, and    -   adding features to the mask whose in-quadrature transmission        component substantially cancels the in-quadrature component of        the transmission discontinuities at edges of the polygonal        elements.

In still yet another aspect of the memory 4300, one or more transmissiondiscontinuities are driven to substantial equality with an allowed valueby:

-   -   giving the desired ratios of spatial frequencies complex values        that provide the image with a desired behavior through focus,    -   forming the 3D topographical representation of the polygon        elements,    -   calculating the transmission discontinuity at the edges of the        polygonal elements, and    -   adding features to the mask whose quadrature transmission        component combined with the quadrature component of the        transmission discontinuities at the edges of the polygonal        elements provides the in-quadrature part of the complex values        of the desired spatial frequency ratios.

Thus it is seen that the foregoing description has provided by way ofexemplary and non-limiting examples a full and informative descriptionof the best apparatus and methods presently contemplated by theinventors for forming lithographic wavefronts. One skilled in the artwill appreciate that the various embodiments described herein can bepracticed individually; in combination with one or more otherembodiments described herein; or in combination with methods andapparatus differing from those described herein. Further, one skilled inthe art will appreciate that the present invention can be practiced byother than the described embodiments; that these described embodimentsare presented for the purposes of illustration and not of limitation;and that the present invention is therefore limited only by the claimswhich follow.

1. A method for generating a desired set of diffracted waves usingfeatures of a lithographic mask for which a set of supportedtransmissions are chosen from a set of supported values, by the stepsof: creating at a computer a representation of the mask as a set ofpolygonal elements, defining constraints for the computer which requirethat the ratio of spatial frequencies in the representation take onamplitude ratios of the desired set of diffracted waves, and using anoptimization algorithm stored in a memory of the computer to adjusttransmission discontinuities at edges of the polygonal elements tosubstantial equality with discontinuity values allowed by the set ofsupported transmissions while maintaining the constraints.
 2. The methodof claim 1 where the optimization algorithm comprises iterated steps,the iterated steps comprising: forming a 3D topographical representationfrom the polygonal elements, and simulating it with a full-3D Maxwellsolver to calculate a Fourier transform of the transmissiondiscontinuities at the edges of the polygonal elements.
 3. The method ofclaim 2 where the iterated steps further comprise: calculating at thecomputer a compensating adjustment that cancels deviations of theFourier transforms of the transmission discontinuities at the edges ofthe polygonal elements from the required spatial frequency ratios. 4.The method of claim 3 where the iterated steps further comprise: formingat the computer an adjusted set of Fourier orders using the compensatededge Fourier transforms calculated in the previous step and use them togenerate with thin-mask wavefront engineering a new set of iteratedpolygonal elements.
 5. The method of claim 4, where the optimizationalgorithm further comprises terminating the iterations when the Fouriertransform of the 3D topographical representation of the iteratedpolygonal elements substantially reproduces the amplitude ratios of thedesired set of diffracted waves.
 6. The method of claim 1 where one ormore transmission discontinuities are driven to substantial equalitywith an allowed value by: forming the 3D topographical representation ofthe polygonal elements, calculating the transmission discontinuity atthe edges of the polygonal elements, and adding features to the maskwhose in-quadrature transmission component substantially cancels thein-quadrature component of the transmission discontinuities at edges ofthe polygonal elements.
 7. The method of claim 1 where one or moretransmission discontinuities are driven to substantial equality with anallowed value by: giving the desired ratios of spatial frequenciescomplex values that provide the image with a desired behavior throughfocus, forming the 3D topographical representation of the polygonelements, calculating the transmission discontinuity at the edges of thepolygonal elements, and adding features to the mask whose quadraturetransmission component combined with the quadrature component of thetransmission discontinuities at the edges of the polygonal elementsprovides the in-quadrature part of the complex values of the desiredspatial frequency ratios.
 8. A memory storing a program of computerreadable instructions executable by a processor to perform actionsdirected to generating a desired set of diffracted waves using featuresof a lithographic mask for which a set of supported transmissions arechosen from a set of supported values, the actions comprising: creatinga representation of the mask as a set of polygonal elements, definingconstraints which require that the ratio of the spatial frequencies inthe representation take on the amplitude ratios of the desired set ofdiffracted waves, and using an optimization algorithm to adjusttransmission discontinuities at edges of the polygonal elements tosubstantial equality with the discontinuity values allowed by the set ofsupported transmissions while maintaining the constraints.
 9. The memoryof claim 8 where the optimization algorithm comprises iterated steps,the iterated steps comprising: forming a 3D topographical representationfrom the polygonal elements, and simulating it with a full-3D Maxwellsolver to calculate a Fourier transform of the transmissiondiscontinuities at the edges of the polygonal elements.
 10. The memoryof claim 9 where the iterated steps further comprise: calculating acompensating adjustment that cancels the deviations of the Fouriertransforms of the transmission discontinuities at the edges of thepolygonal elements from the required spatial frequency ratios.
 11. Thememory of claim 10 where the iterated steps further comprise: forming anadjusted set of Fourier orders using the compensated edge Fouriertransforms calculated in the previous step and use them to generate withthin-mask wavefront engineering a new set of iterated polygonalelements.
 12. The memory of claim 11, where the optimization algorithmfurther comprises terminating the iterations when the Fourier transformof the 3D topographical representation of the iterated polygonalelements substantially reproduces the amplitude ratios of the desiredset of diffracted waves.
 13. The memory of claim 8 where one or moretransmission discontinuities are driven to substantial equality with anallowed value by: forming the 3D topographical representation of thepolygonal elements, calculating the transmission discontinuity at theedges of the polygonal elements, and adding features to the mask whosein-quadrature transmission component substantially cancels thein-quadrature component of the transmission discontinuities at edges ofthe polygonal elements.
 14. The memory of claim 8 where one or moretransmission discontinuities are driven to substantial equality with anallowed value by: giving the desired ratios of spatial frequenciescomplex values that provide the image with a desired behavior throughfocus, forming the 3D topographical representation of the polygonelements, calculating the transmission discontinuity at the edges of thepolygonal elements, and adding features to the mask whose quadraturetransmission component combined with the quadrature component of thetransmission discontinuities at the edges of the polygonal elementsprovides the in-quadrature part of the complex values of the desiredspatial frequency ratios.